Edge cracked piezoelectric ceramic block under electromechanical impact loading

The dynamic field intensity factors and energy release rates in a piezoelectric ceramic block containing an edge crack with the condition of continuous electric crack faces under electromechanical impact loading are obtained. Integral transform method is used to reduce the problem to two pairs of dual integral equations, which are then expressed to an Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor and dynamic energy release rate are obtained to show the influence of the geometry and electric field.     

SrTiO3材料中氧空位的密度泛函理论

计算了SrTiO3-δ（δ=0，δ=0．125）体系电子结构，分析了氧空位对SrTiO3晶体的价键结构、能带、态密度、分波态密度、差分电荷密度的影响。所有计算都是基于密度泛函理论（DFT）框架下的第一性原理平面波超软赝势方法。计算结果表明：当氧空位浓度δ=0．125时，空位在母体化合物SrTiO3中引入了大量的传导电子，费米能级进入导带，体系显示金属型导电性。由于空位掺杂，导带底附近的态密度发生了畸变，刚性能带模型不再适合描述SrTiO2.875体系。同时，在导带底附近距离费米能级0．3eV处引入了空位能级，这和实验测得的SrTiO3材料内中性氧空位的电离能约为0．4eV相符。此外，Mulliken布局分析、分波态密度和差分电荷密度分析表明，该空位能级主要由与其最近邻的两个Ti原子的3d电子态贡献，并且由该空位引入的导电电子大部分都局域在空位最近邻的两个Ti原子周围。最后，计算了三种典型平衡条件下SrTiO3晶体内中性氧空位的形成能。     

Describing metal surfaces and nanostructures with orbital-free density functional theory

Orbital-free density functional theory (OF-DFT) can be made to scale linearly with sample size, allowing thousands of atoms to be treated explicitly with quantum mechanics. State-of-the-art kinetic energy density functionals and ion–electron pseudopotentials are used to obtain accurate structural property predictions for nanoparticles, nanowires, extended surfaces, and nanoindentation of simple metals.     

Commutativity of the strain energy density expression for the benefit of the FEM implementation of Koiter's initial postbuckling theory

The concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature.     

基于虚拟裂纹闭合技术的应变能释放率分析

基于虚拟裂纹闭合技术(VCCT),建立了复合材料层合板层间裂纹尖端的应变能释放率(SERR)三维有限元计算模型。该模型考虑了裂纹尖端大转动和离散单元形状变化对应变能释放率计算的影响,修正了裂纹尖端应变能释放率的计算方法。利用该模型计算了裂纹长度为15 mm和35 mm时纯Ⅰ型和纯Ⅱ型的应变能释放率,纯Ⅰ型应变能释放率分别为 207 J/m2和 253 J/m2;纯Ⅱ型应变能释放率分别为 758 J / m 2和 1040 J / m2;计算值与试验值吻合得很好。同时,该模型计算了混合型不同比值 R=(G_Ⅱ/G_Ⅰ+G_Ⅱ)的长裂纹层合板层间断裂过程的应变能释放率,其中Ⅰ型和Ⅱ型应变能释放率计算值与试验平均值的最大误差为 11.4%,最小误差为 0.4%。该模型能有效计算裂纹尖端的应变能释放率。     

Virtual crack extension method for calculating the second order derivatives of energy release rates for multiply cracked systems

In this paper, we further generalize the work of Lin and Abel [Lin SC, Abel JF. Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 1988;38:217-35.] to the case of higher order derivatives of energy release rates for two-dimensional, multiply cracked systems. The direct integral expressions are presented for the energy release rates and their first and second order derivatives. The salient feature of this numerical method is that the energy release rates and their first and second order derivatives can be computed in a single analysis. It is demonstrated through a set of examples that the proposed method gives expectedly decreasing, but acceptably accurate results for the energy release rates and their first and second order derivatives. The computed errors were approximately 0.5% for the energy release rates, 3-5% for their first order derivatives and 10-20% for their second order derivatives for the mesh densities used in the examples. Potential applications of the present method include a universal size effect model and a probabilistic fracture analysis of cracked structures.     

Comparison of virtual crack extension and strain energy density methods applied to contact surface crack growth

The paper is concerned with comparison of two crack propagation methods applied to a two-dimensional computational model of the surface initiated crack growth in the lubricated contact area of meshing gears. The virtual crack extension method and the minimum strain energy density criterion are used for simulation of the crack propagation in the framework of the finite element analysis. The discretised equivalent contact model, with the assumed size and orientation of the initial crack, is subjected to contact loading conditions, accounting for the elasto-hydro-dynamic lubrication effects, tangential loading due to sliding and the influence of lubricating fluid, driven into the crack by hydraulic mechanism. The computational results show that both crack propagation methods give comparable results, although the virtual crack extension method has some clear advantages due to its theoretical superiority in dealing with mixed-mode short crack propagation close to the loading boundary.     

Analytical wide-range weight functions for various finite cracked bodies

Closed-form wide-range weight functions have been presented for various finite plane cracked bodies. A unified analytical procedure was used in the derivation. First, accurate crack face displacement expressions for center and edge cracks were determined for the polynomial type reference load case. These displacements were then used to derive analytical weight functions, whose accuracy was critically assessed using the related Green's functions. Stress intensity factors formulae for a number of basic load cases including concentrated forces, polynomial as well as a band of linearly varying stress, have been obtained. These basic solutions combined with superposition method enable stress intensity factors to be rapidly determined for complex loadings, as demonstrated by example engineering crack problems. Discussions were made on the reference load case dependence of the weight functions, and the significance of the number of terms contained in the crack face displacement representation on the solution accuracy at extended crack lengths. The analytical wide-range weight functions have been proved versatile, very cost-saving, easy-to-use, and accurate.     

Barnacles resist removal by crack trapping

We study the mechanics of pull-off of a barnacle adhering to a thin elastic layer which is bonded to a rigid substrate. We address the case of barnacles having acorn shell geometry and hard, calcarious base plates. Pull-off is initiated by the propagation of an interface edge crack between the base plate and the layer. We compute the energy release rate of this crack as it grows along the interface using a finite element method. We also develop an approximate analytical model to interpret our numerical results and to give a closed-form expression for the energy release rate. Our result shows that the resistance of barnacles to interfacial failure arises from a crack-trapping mechanism.     

Determination of energy release rate and mode mix in three-dimensional layered structures using plate theory

A plate theory-based method for determining energy release rates is presented for general loadings of three dimensional layered structures. Mode decomposition is performed for cases that exhibit an inverse square root singularity and for which certain other restrictions apply. Predictions for energy release rate and mode mix for typical problems are presented and verified by comparison with results obtained by three dimensional finite element analyses.     

Off-axis fatigue crack growth and the associated energy release rate in composite laminates

Stable matrix crack growth behaviour under mechanical fatigue loading has been studied in a quasi-isotropic (0/90/-45/+45)_s GFRP laminate. Detailed experimental observations were made on the accumulation of cracks and on the growth of individual cracks in +45° as well as 90° plies. A generalised plain strain finite element model of the damaged laminate has been constructed. This model has been used to relate the energy release rate of growing cracks to the crack growth rate via a Paris relation.     

用应变能密度法分析纤维金属层板的微动疲劳特性

为研究纤维金属层板的微动疲劳特性,首先,基于三维坐标系下的临界平面法求解了纤维金属层板铝层临界平面上的应力和应变分量,并进一步求解了Smith-Watson-Topper （SWT）和I型Nita-Ogatta-Kuwabara （NOK）应变能密度参数;然后,建立了应变能密度参数-微动疲劳寿命关系式,并通过实验数据得到了寿命预测公式中的待定参数;最后,采用I型NOK应变能密度准则分析了铝层厚度、纤维层厚度、各层相对厚度和桥足圆角半径等对微动疲劳损伤位置和寿命的影响,并为纤维金属层板抗微动疲劳设计提出了一些合理化建议。结果表明:增加铝层厚度可以延长微动疲劳寿命,但增加纤维层厚度和桥足圆角半径不会改善微动疲劳特性。提出的方法可为分析纤维金属层板铆接和螺栓连接中的微动疲劳问题提供理论依据。      

二元合金自由能的统计理论

为了处理二元合金中多种相的相平衡和相变总是根据准化学近似法和熵的统计理论,提出了二元合金自由能统计理论的一般处理模式,推导了自由能与成分和有序参数关系的表达式,由此对Ｆｅ－Ｃｏ、Ｐｔ－Ｃｏ和Ｒ－Ｃｏ合金的相平衡和相变问题进行了处理,获得了理想的结果。     

Reconsidering the Paradox of Rice for a Linear Strain Hardening Material

Rice (1966) considered a growing crack in an elastic-plastic material where the flow strength saturates, and demonstrated that there is no energy available for the propagation. This is the famous «Paradox of Rice». Kfouri and Miller (1976) confirmed this result using an elastic-plastic finite element solution for an linear strain hardening material. We show in this paper that the result of Kfouri and Miller is questionable.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Evaluation of the strain energy density control volume for a nanoscale singular stress field

Fracture mechanics at micro‐ and nano‐scale has become a very attractive topic in the last years. However, the results are still few, mostly because of the lack of effective analytical tools and of the difficult to conduct experimental tests at those scales. In this study, the authors report preliminary analysis on the application of the Strain Energy Density (SED) method at nano‐scale. In detail, starting from mechanical properties experimentally evaluated on small single crystal silicon cracked specimens, a first evaluation of the control volume due to a nano‐size singular stress field is carried out. If the extension of the SED approach at micro‐ nano‐scale is given in near future, an easy and fast tool to design against fatigue will be provided for micro‐ nano‐devices such as MEMS and NEMS, resulting in a significant technological impact and providing an easy and fast tool to conduct static and fatigue assessment at micro‐ and nano‐scale.     

Averaged strain energy density estimated rapidly from nodal displacements by coarse FE analyses: Cracks under mixed mode loadings

The strain energy density (SED) averaged over a structural volume allows to assess the fracture and fatigue behaviour of cracked components under mixed‐mode loadings. To rapidly estimate the averaged SED, two approaches have been previously proposed: (a) the direct approach; (b) the peak stress method (PSM) and nodal stress (NS) approach. In this paper, the nodal displacement (ND) approach is presented to rapidly estimate the averaged SED from the nodal displacements by FE analyses. This method combines the advantages of all previous approaches, ie, the use of coarse meshes able to capture the contributions of both stress intensity factors (SIFs) and higher order terms, without requiring geometrical modelling of the control volume. To validate the method, cracked plates and cracked bars under mixed‐mode loadings have been analysed. The averaged SED values estimated by the ND approach agree with those calculated by the direct approach, within a range of applicability.     

Applying strain energy density factor theory to propagation estimating of surface crack in tubular T-joints

The propagation direction of a surface crack in tubular T-joints and the surface crack profile are studied in this paper. By using Photoelasticity and Finite Element method, the distribution of Stress Intensity Factor (SIF) along the front edge of the prefabricated semi-elliptic surface cracks at the hot spots of tubular T-joints is obtained. The results show that the cracks are subjected to mix-mode loading, and the distribution of SIF along the front edge of the crack depends not only on a/T, the ratio of crack depth to the thickness of chord wall, but also largely on shape ratio a/c, the ratio of depth of the crack a to its length c. It can also be predicted that there exists a certain precedent propagation direction for a surface crack with a certain a/c. According to the Strain Energy Density Factor theory the condition of constant strain energy density factor S on the front edge of the crack will be enforced on each point of the prosective crack profile, so that the crack profile can be pre-estimated.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.     

On the dynamic energy release rate in functionally graded materials

By using the effective shear modulus and mass density, the influence of functional gradient on dynamic energy release rate is discussed under the condition of constant velocity of crack propagation.